Revise Scalars and vectors for Cambridge 9702 (1.4)

A scalar quantity is defined solely by its magnitude – essentially, its size or amount. Scalars have no associated direction. Think of how you measure your mass on a scale; it's just '70 kg', not '70 kg upwards'. These quantities simplify many everyday measurements, focusing only on 'how much'.

In contrast, a vector quantity requires both a magnitude and a specific direction to be fully described. For instance, when you talk about your displacement, it's not just '5 meters' but '5 meters North'. Vectors are often represented visually as an arrow; the arrow's length corresponds to the vector's magnitude, and the arrowhead points in its direction.

When two vectors act at right angles (perpendicular) to each other, combining them to find their resultant vector (the single vector equivalent to their combined effect) is straightforward. Imagine two forces, one pulling right and one pulling up. The overall effect, the resultant, can be found using fundamental geometry.

Resultant Magnitude (Pythagoras' Theorem): $R = \sqrt{V_x^2 + V_y^2}$

Resultant Direction (Trigonometry): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_y}{V_x}$

Sometimes, we need to do the reverse of adding: break down a single vector into two components that are at right angles to each other. This process is called vector resolution. It's incredibly useful when a force or velocity acts at an angle, and you need to see its individual effects along, say, horizontal and vertical axes.

If a vector $V$ makes an angle $\theta$ with the horizontal (x-axis): Component adjacent to the angle: $V_x = V \cos \theta$ Component opposite to the angle: $V_y = V \sin \theta$

When vectors aren't at right angles, you can't use Pythagoras directly. One common and accurate method is to first resolve each vector into its perpendicular components (e.g., horizontal and vertical). Once you have all the horizontal components and all the vertical components, you can add them algebraically to get total horizontal ($R_x$) and total vertical ($R_y$) resultants. Then, combine these two total perpendicular components using Pythagoras' theorem and trigonometry to find the final resultant.

Alternatively, for a more visual approach, you can use scale diagrams and the head-to-tail method. Draw the first vector to scale, then draw the second vector (also to scale) starting from the head (tip) of the first. The resultant is the vector drawn from the tail (start) of the first to the head of the last.

Multiplying a vector by a scalar changes its magnitude. If you multiply a vector by a positive scalar, its magnitude changes proportionally, but its direction remains the same. For example, $2\mathbf{v}$ is a vector twice as long as $\mathbf{v}$ but in the identical direction.

If you multiply a vector by a negative scalar, its magnitude also changes proportionally, but its direction is reversed. So, $-3\mathbf{v}$ would be a vector three times the length of $\mathbf{v}$ but pointing in precisely the opposite direction.